In the theory of
dynamical systems
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models ...
and
control theory
Control theory is a field of control engineering and applied mathematics that deals with the control system, control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the applic ...
, a
linear
In mathematics, the term ''linear'' is used in two distinct senses for two different properties:
* linearity of a '' function'' (or '' mapping'');
* linearity of a '' polynomial''.
An example of a linear function is the function defined by f(x) ...
time-invariant system
In control theory, a time-invariant (TI) system has a time-dependent system function that is not a direct function of time. Such systems are regarded as a class of systems in the field of system analysis. The time-dependent system function is a ...
is marginally stable if it is neither
asymptotically stable nor
unstable
In dynamical systems instability means that some of the outputs or internal state (controls), states increase with time, without bounds. Not all systems that are not Stability theory, stable are unstable; systems can also be marginal stability ...
. Roughly speaking, a system is stable if it always returns to and stays near a particular state (called the
steady state
In systems theory, a system or a process is in a steady state if the variables (called state variables) which define the behavior of the system or the process are unchanging in time. In continuous time, this means that for those properties ''p' ...
), and is unstable if it goes further and further away from any state, without being bounded. A marginal system, sometimes referred to as having neutral stability,
is between these two types: when displaced, it does not return to near a common steady state, nor does it go away from where it started without limit.
Marginal stability, like instability, is a feature that control theory seeks to avoid; we wish that, when perturbed by some external force, a system will return to a desired state. This necessitates the use of appropriately designed control algorithms.
In
econometrics
Econometrics is an application of statistical methods to economic data in order to give empirical content to economic relationships. M. Hashem Pesaran (1987). "Econometrics", '' The New Palgrave: A Dictionary of Economics'', v. 2, p. 8 p. 8 ...
, the presence of a
unit root
In probability theory and statistics, a unit root is a feature of some stochastic processes (such as random walks) that can cause problems in statistical inference involving time series models. A linear stochastic process has a unit root if ...
in observed
time series
In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. ...
, rendering them marginally stable, can lead to invalid
regression results regarding effects of the
independent variable
A variable is considered dependent if it depends on (or is hypothesized to depend on) an independent variable. Dependent variables are studied under the supposition or demand that they depend, by some law or rule (e.g., by a mathematical function ...
s upon a
dependent variable
A variable is considered dependent if it depends on (or is hypothesized to depend on) an independent variable. Dependent variables are studied under the supposition or demand that they depend, by some law or rule (e.g., by a mathematical functio ...
, unless appropriate techniques are used to convert the system to a stable system.
Continuous time
A
homogeneous
Homogeneity and heterogeneity are concepts relating to the uniformity of a substance, process or image. A homogeneous feature is uniform in composition or character (i.e., color, shape, size, weight, height, distribution, texture, language, i ...
continuous linear time-invariant system
In system analysis, among other fields of study, a linear time-invariant (LTI) system is a system that produces an output signal from any input signal subject to the constraints of Linear system#Definition, linearity and Time-invariant system, ...
is marginally stable
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
the real part of every
pole (
eigenvalue
In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...
) in the system's
transfer-function is
non-positive, one or more poles have zero real part, and all poles with zero real part are
simple root
In mathematics, a polynomial is a mathematical expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication and exponentiation to nonnegative integer ...
s (i.e. the poles on the
imaginary axis are all distinct from one another). In contrast, if all the poles have strictly negative real parts, the system is instead asymptotically stable. If the system is neither stable nor marginally stable, it is unstable.
If the system is in
state space representation
State most commonly refers to:
* State (polity), a centralized political organization that regulates law and society within a territory
**Sovereign state, a sovereign polity in international law, commonly referred to as a country
**Nation state, a ...
, marginal stability can be analyzed by deriving the
Jordan normal form
\begin
\lambda_1 1\hphantom\hphantom\\
\hphantom\lambda_1 1\hphantom\\
\hphantom\lambda_1\hphantom\\
\hphantom\lambda_2 1\hphantom\hphantom\\
\hphantom\hphantom\lambda_2\hphantom\\
\hphantom\lambda_3\hphantom\\
\hphantom\ddots\hphantom\\
...
: if and only if the Jordan blocks corresponding to poles with zero real part are scalar is the system marginally stable.
Discrete time
A homogeneous
discrete time
In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which variables that evolve over time are modeled.
Discrete time
Discrete time views values of variables as occurring at distinct, separate "poi ...
linear time-invariant system is marginally stable if and only if the greatest magnitude of any of the poles (eigenvalues) of the transfer function is 1, and the poles with magnitude equal to 1 are all distinct. That is, the transfer function's
spectral radius
''Spectral'' is a 2016 Hungarian-American military science fiction action film co-written and directed by Nic Mathieu. Written with Ian Fried (screenwriter), Ian Fried & George Nolfi, the film stars James Badge Dale as DARPA research scientist Ma ...
is 1. If the spectral radius is less than 1, the system is instead asymptotically stable.
A simple example involves a single first-order
linear difference equation
In mathematics (including combinatorics, linear algebra, and dynamical systems), a linear recurrence with constant coefficients (also known as a linear recurrence relation or linear difference equation) sets equal to 0 a polynomial that is linear ...
: Suppose a state variable ''x'' evolves according to
:
with parameter ''a'' > 0. If the system is perturbed to the value
its subsequent sequence of values is
If ''a'' < 1, these numbers get closer and closer to 0 regardless of the starting value
while if ''a'' > 1 the numbers get larger and larger without bound. But if ''a'' = 1, the numbers do neither of these: instead, all future values of ''x'' equal the value
Thus the case ''a'' = 1 exhibits marginal stability.
System response
A marginally stable system is one that, if given an
impulse of finite magnitude as input, will not "blow up" and give an unbounded output, but neither will the output return to zero. A bounded offset or oscillations in the output will persist indefinitely, and so there will in general be no final steady-state output. If a continuous system is given an input at a frequency equal to the frequency of a pole with zero real part, the system's output will increase indefinitely (this is known as pure resonance
). This explains why for a system to be
BIBO stable, the real parts of the poles have to be strictly negative (and not just non-positive).
A continuous system having imaginary poles, i.e. having zero real part in the pole(s), will produce sustained oscillations in the output. For example, an undamped second-order system such as the suspension system in an automobile (a
mass–spring–damper system), from which the damper has been removed and spring is ideal, i.e. no friction is there, will in theory oscillate forever once disturbed. Another example is a frictionless
pendulum
A pendulum is a device made of a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate i ...
. A system with a pole at the origin is also marginally stable but in this case there will be no oscillation in the response as the imaginary part is also zero (''jw'' = 0 means ''w'' = 0 rad/sec). An example of such a system is a mass on a surface with friction. When a sidewards impulse is applied, the mass will move and never returns to zero. The mass will come to rest due to friction however, and the sidewards movement will remain bounded.
Since the locations of the marginal poles must be ''exactly'' on the imaginary axis or unit circle (for continuous time and discrete time systems respectively) for a system to be marginally stable, this situation is unlikely to occur in practice unless marginal stability is an inherent theoretical feature of the system.
Stochastic dynamics
Marginal stability is also an important concept in the context of
stochastic dynamics
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables in a probability space, where the index of the family often has the interpretation of time. Stoc ...
. For example, some processes may follow a
random walk
In mathematics, a random walk, sometimes known as a drunkard's walk, is a stochastic process that describes a path that consists of a succession of random steps on some Space (mathematics), mathematical space.
An elementary example of a rand ...
, given in discrete time as
:
where
is an
i.i.d. error term In mathematics and statistics, an error term is an additive type of error.
In writing, an error term is an instance of faulty language or grammar.
Common examples include:
* errors and residuals in statistics, e.g. in linear regression
* the error ...
. This equation has a
unit root
In probability theory and statistics, a unit root is a feature of some stochastic processes (such as random walks) that can cause problems in statistical inference involving time series models. A linear stochastic process has a unit root if ...
(a value of 1 for the eigenvalue of its
characteristic equation), and hence exhibits marginal stability, so special
time series
In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. ...
techniques must be used in empirically modeling a system containing such an equation.
Marginally stable
Markov processes Markov (Bulgarian language, Bulgarian, ), Markova, and Markoff are common surnames used in Russia and Bulgaria. Notable people with the name include:
Academics
*Ivana Markova (1938–2024), Czechoslovak-British emeritus professor of psychology at t ...
are those that possess
null recurrent classes.
See also
*
Lyapunov stability
Various types of stability may be discussed for the solutions of differential equations or difference equations describing dynamical systems. The most important type is that concerning the stability of solutions near to a point of equilibrium. ...
*
Exponential stability
In control theory, a continuous linear time-invariant system (LTI) is exponentially stable if and only if the system has eigenvalues (i.e., the poles of input-to-output systems) with strictly negative real parts (i.e., in the left half of the c ...
References
{{DEFAULTSORT:Marginal Stability
Dynamical systems
Stability theory